1. Field of the Invention
The present invention relates to a surface shape calculation method and apparatus.
2. Description of the Related Art
A stitching method or synthetic aperture method using an interferometer is known as a method of measuring the surface shape (planar shape) of an optical element such as a large-diameter mirror or lens. In the stitching method using an interferometer, it is important to separate the system error of the interferometer from the surface shape of the target surface (R. Mercier et. al., “Two-flat method for bi-dimensional measurement of absolute departure from the best sphere”, Appl. Opt., 6 (1997), 117 (“Reference 1”). Reference 1 discloses a technique of measuring the surface shape of a target surface while relatively shifting the target surface and the reference surface, and separating a system error from the surface shape of the target surface based on measured values at portions where the measured values before and after the shift overlap.
U.S. Pat. No. 6,956,657 (“Reference 2”) discloses a technique of measuring the surface shapes of a plurality of regions (sub aperture regions) each smaller than the whole target surface and combining the measured values of the plurality of regions, thereby obtaining the surface shape of the target surface. Note that in Reference 2, the system error is separated from the target surface shape based on measured values of portions where the plurality of regions overlap, as in Reference 1. More specifically, the measured value of the j-th region is represented by the sum of three elements, that is, the surface shape of the target surface, the alignment error between the interferometer and the target surface, and the system error. Then, the alignment error and the system error are obtained by the least squares method such that the difference between the measured values of portions where adjacent regions overlap is minimized.
Note that in the conventional techniques of references 1 and 2, the system error is expressed by the sum of polynomials such as Zernike polynomials or trigonometric functions.
However, since the conventional techniques use the stitching method that inputs the measured value of each of the plurality of regions of the target surface, an error is generated in the surface shape of the target surface to be measured due to restrictions on the number of terms to be used in polynomials for expressing the system error. That is, since a system error that cannot be expressed by the number of terms to be used in polynomials remains as an error, the accuracy of measuring the surface shape of the target surface is reduced.
This problem can be avoided by sufficiently increasing the number of terms of polynomials for expressing the system error. In general, however, if the number of terms of polynomials that are fitting variables increases, the matrix to be used in the least squares method becomes large. This leads to a longer calculation time or shortage of computer memory capacity. Hence, in the conventional techniques, the number of terms to be used in polynomials for expressing a system error needs to be limited, and it is therefore difficult to obtain a sufficient measurement accuracy.